發(fā)布時(shí)間：2025-02-06 14:16

　　界面問(wèn)題廣泛存在于實(shí)際應(yīng)用中,如流體力學(xué),電磁波的傳播、材料科學(xué)和生物科學(xué)。它通常涉及求解耦合的偏微分方程組。本文致力于研究界面問(wèn)題的有限元方法。根據(jù)網(wǎng)格單元和界面之間的拓?fù)潢P(guān)系,界面問(wèn)題的有限元方法(FEMs)可分為兩大類(lèi),即界面匹配網(wǎng)格方法和界面非匹配網(wǎng)格方法。界面匹配網(wǎng)格方法的優(yōu)點(diǎn)在于誤差分析簡(jiǎn)單,并且收斂階是最優(yōu)的。然而,在界面隨時(shí)間演變的情況下,讓網(wǎng)格匹配界面需要重新剖分網(wǎng)格。當(dāng)界面拓?fù)浣Y(jié)構(gòu)變化時(shí),例如破裂或者合并,生成匹配界面的網(wǎng)格是很困難的。因此,界面非匹配網(wǎng)格方法成了一個(gè)重要的研究方向。界面非匹配網(wǎng)格方法主要有兩種,擴(kuò)展有限元法(XFEMs)和浸入界面有限元方法(IFEMs)。兩種方法都是對(duì)有限元空間進(jìn)行修正,以得到最優(yōu)的插值誤差估計(jì)。但是,這兩種方法都有各自的一些缺點(diǎn)。擴(kuò)展有限元方法有許多不同的種類(lèi),其中只有尼采-擴(kuò)展有限元方法有嚴(yán)格的理論分析。對(duì)于尼采-擴(kuò)展有限元方法,它破壞了解的連續(xù)性,因此需要在離散的弱形式加額外的懲罰項(xiàng)。對(duì)于浸入界面有限元方法,它的基函數(shù)構(gòu)造依賴(lài)于界面跳躍條件并且誤差分析相對(duì)困難。為了克服這些缺點(diǎn),我們提出了一種新的界面非匹配網(wǎng)格方法,即協(xié)調(diào)增擴(kuò)...

【文章頁(yè)數(shù)】：102 頁(yè)

【學(xué)位級(jí)別】：博士

【文章目錄】：
摘要
Abstract
Chapter 1 Introduction
    1.1 Model problems and applications
    1.2 An overview of FEMs for interface problems
    1.3 Notation and Definitions
Chapter 2 A conforming enriched finite element method for elliptic inter-face problems
    2.1 The conforming enriched finite element method
    2.2 Properties of the enrichment function
    2.3 Error analysis
    2.4 Numerical examples
        2.4.1 Numerical examples with two sub-domains
        2.4.2 Numerical examples with three sub-domains
        2.4.3 Numerical examples with variable coefficients
Chapter 3 A conforming enriched finite element method for Stokes inter-face problems
    3.1 Weak forms of Stokes interface problems
    3.2 Stability analysis
    3.3 Error analysis
        3.3.1 Approximation properties
        3.3.2 An a prior error estimate
    3.4 Numerical examples
        3.4.1 Example 1: the case of a piecewise constant viscosity
        3.4.2 Example 2: the case of a variable viscosity
Chapter 4 A conforming enriched finite element method for Stokes-ellipticinterface problems
    4.1 Weak forms of Stokes-elliptic interface problems
    4.2 Stability analysis
    4.3 Error analysis
        4.3.1 Approximation properties
        4.3.2 An a prior error estimate
    4.4 Numerical examples
        4.4.1 Example 1: the case of a piecewise constant viscosity
        4.4.2 Example 2: the case of a variable viscosity
Chapter 5 Conclusions and future works
    5.1 A framework of FEMs for interface problems
    5.2 Future works
Bibliography
Publications and Completed Papers
Acknowledgements



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